Aaron Hernandez, who has pleaded not guilty to the Lloyd murder charges, is being held without bail at Bristol County House of Correction.(John Tlumacki/Globe Staff/File)

Prosecutors have begun making the case to a Suffolk County grand jury that former New England Patriot Aaron Hernandez should be charged in the shooting death of two men on a South End street last July, according to two law enforcement officials briefed on the investigation.

“The case against Hernandez appears to be strengthening,” one of the officials said. They spoke on condition of anonymity because they are not authorized to discuss the case publicly.

The proceedings are conducted behind closed doors, so details of the case against Hernandez are unknown. But the fact that the case has gone to a grand jury indicates that the investigation of the killing of two Cape Verdean immigrants is now more narrowly focused on Hernandez. Safiro Furtado and Daniel Abreu were shot early on July 16, 2012, in the South End when someone in a sport utility vehicle with Rhode Island license plates pulled up next to their car and began firing.

Hernandez’s lawyer, Michael Fee, did not return e-mails or messages left for him at his office.

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The grand jury proceedings also raise the troubling prospect that Hernandez, a talented tight end for the Patriots, might have been playing football last season after he had killed two people.

Boston police made a connection between Hernandez and the South End double murder after he came under scrutiny in the June 17 killing of Odin Lloyd, a 27-year-old Dorchester man.

Hernandez, who has pleaded not guilty to the Lloyd murder charges, is being held without bail at Bristol County House of Correction; he is due Wednesday in Attleboro District Court for a probable cause hearing.

Law enforcement officials have told the Globe that investigators believe Hernandez drove Lloyd to an industrial park in North Attleborough, then shot him because he was worried Lloyd had information about the South End double homicide and might talk.

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During the drive to North Attleborough, Hernandez told Lloyd he was “ ‘chilling’ with people that he had problems with,” according to an affidavit unsealed Tuesday in Bristol, Conn. The affidavit said investigators learned this from Carlos Ortiz, one of the two men with Hernandez the night Lloyd was shot.

Boston police declined to comment on the Suffolk grand jury, as did Suffolk District Attorney Daniel F. Conley, whose office is overseeing the investigation.

“The grand jury proceedings are by their nature confidential, and we do not comment on them or on open investigations,” said Jake Wark, Conley’s spokesman.

One of the officials with knowledge of the Suffolk grand jury said that investigators are not in a hurry to charge Hernandez with the killings of Furtado and Abreu because he is being held without bail.

The motive behind the killings of Furtado and Abreu, friends who worked together for a cleaning company, remains unknown. Their relatives have said the men did not know Hernandez and neither man had a criminal record.

Furtado’s mother, Maria Teixiera, 59, said it is difficult for her to feel hopeful that her son’s case could be progressing.

“I just want to know for certain that the person who did this has been caught. Until then, what is there for me to say?” she said, speaking in Creole, as she stood in the foyer of her Dorchester home, surrounded by her grandchildren. “I just want justice for him. If the police are close to finding out who killed him . . . well, I just want them to tell me when they actually catch the person. That’s what my family waits for.”

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In Massachusetts, all those charged with murder must be indicted by a grand jury before they can be placed on trial. Often that happens after a suspect is arrested and arraigned in the district court system.

The grand jury is made up of 23 people picked from the same pool used to select jurors for public trials. Unlike those trials, however, there is no judge or defense attorneys. Grand jurors listen only to a prosecutor and any witness who receives a subpoena to testify. A grand jury in Suffolk will sit for three to four days a week for three to four months hearing evidence in multiple cases.

The grand jury hearing the Hernandez case is also hearing evidence in other, unrelated felony cases.

Grand jurors do not need to be convinced beyond a reasonable doubt that a defendant is guilty. They must only decide, by majority vote, whether there is enough probable cause to indict a defendant.

“It’s a very one-sided presentation,” said Michael Cassidy, a professor at Boston College Law School and former chief of the criminal bureau in the Massachusetts Attorney General’s Office. “The standard is probable cause, which is a fairly low standard. . . . There is an expression in law enforcement that a grand jury would indict a ham sandwich if the government asked it to.”

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For the past several years, prosecutors have also used the grand jury process to strengthen their cases by obtaining evidence that would be harder to come by in a traditional investigation.

They can subpoena reluctant witnesses who would otherwise rebuff a detective who approached them on the street. Prosecutors can also use the grand jury process to subpoena evidence like the suspect’s DNA, or bank or phone records.

The grand jury process has helped prosecutors bring more solid cases to trial, said Stephen Weymouth, a defense attorney who has represented numerous defendants in Suffolk County.

“I think that the nature of the investigations and the quality of the cases has improved tremendously over the last decade,” he said.

In other developments in the Hernandez investigation Tuesday, a Connecticut judge unsealed three affidavits that were the basis for three search warrant applications connected to the Lloyd killing. Police searched the home of Hernandez’s uncle in Bristol, Hernandez’s hometown.

Authorities became interested in that address after Ortiz told investigators that he drove back to the house after Lloyd’s killing, accompanied by Ernest Wallace, the other man who was with Hernandez the night Lloyd was killed.

It was there that Boston police eventually tracked down the SUV that investigators believe was used in the South End double killing last year.

Wesley Lowery and Brian Ballou of the Globe Staff contributed to this report. Maria Cramer can be reached at mcramer@globe.com. Follow her on Twitter @GlobeMCramer.